Regular polygon detection
Nick Barnes1 , Gareth Loy2 , David Shaw1 , Antonio Robles-Kelly1
1. National ICT Australia, Locked Bag 8001, Canberra, ACT, 2601, Australia,
Dept. Information Engineering, The Australian National University.
2. Computer Vision & Active Perception Laboratory, Royal Institute of Technology (KTH), Stockholm
Abstract
feature detection. Grouping approaches take low-level features, such as edges (but also alternatively raw edge pixels)
and apply matching and clustering based methods.
In perceptual grouping, image elements are grouped according to some perceptual criteria. Several authors have
investigated iterative relaxation style operators for grouping edgels. This began with Shashua and Ullman [19],
which required many iterations. This was refined by Guy
and Medioni [7] among others so as to be O(k 2 ) where k is
the number of edgels. At a high level this problem can be
composed as recovering a graph of relational arrangement.
A common approach is spectral graph theory, e.g., [16, 18].
Probabilistic approaches in this area include Bayes nets [5]
and combining evidence from raw edge attributes [3]. More
recent approaches have used EM, e.g., [1, 4]. One of the basic algorithmic approaches here is examining pairwise relations. If the elements being examined is edge pixels then the
complexity is at least O(N 2 ), where N is the image size.
Alternatively, we may fit functions to the image data.
The Hough transform [8] and circular Hough transform [15]
vote to an in-place representation of the shape to be detected. The circular Hough detects circle centres, and the
Hough transform detects the closest line point to the origin
(this mapping is in the same space in the log-Hough transform [21] in log-polar images). Each pixel ‘votes’ for each
feature that it could possibly be a part of. This approach
is inherently robust, as gaps, noise, and partial occlusion
are ignored, but appear as a decreased strength of the feature. Such algorithms are local in their computation with
each point only being registered for shapes that it can be
part of. These methods have been generalised [2] to detect
arbitrary shapes. The original Hough transform has a relatively simple mapping for each pixel, however, the generalisations vote into higher dimensions, and quickly become
computationally intensive on serial architectures. Further,
parallelisation is non-trivial due to the high dimensionality.
The Hough transform can be formulated as maximum
likelihood estimation [20]. Kiryati and Bruckstein [10] formulated robust maximum likelihood line finding on a grid,
assuming independent noise between points. Geyer et al.
[6], applied a sampled likelihood function approach to estimation of the essential matrix. Makadia and Daniilidis [14]
formulated a Hough-like algorithm in motion space for estimating robot position using a Radon transform formulation
This paper describes a new robust regular polygon detector. The regular polygon transform is posed as a mixture
of regular polygons in a five dimensional space. Given the
edge structure of an image, we derive the a posteriori probability for a mixture of regular polygons, and thus the probability density function for the appearance of a mixture of
regular polygons. Likely regular polygons can be isolated
quickly by discretising and collapsing the search space into
three dimensions. The remaining dimensions may be efficiently recovered subsequently using maximum likelihood
at the locations of the most likely polygons in the subspace.
This leads to an efficient algorithm. Also, the a posteriori
formulation facilitates inclusion of additional a priori information leading to real-time application to road sign detection. The use of edge orientation information also reduces
noise compared to existing approaches such as the generalised Hough transform. Results are presented for images
with noise to show stability. Also, the detector is applied to
two separate applications: real-time road sign detection for
on-line driver assistance; and, feature detection, recovering
stable features in rectilinear environments.
1
Introduction
In scenes containing manufactured artefacts, features often appear as regular polygons. In road scenes, triangular,
rectangular and octagonal signs display critical information.
Polygonal shapes appear frequently in the structure of office environment. Corners are common partial polygons,
but over a large range of scales regular polygons represent
much more indoor structure. In outdoor scenes, brickwork,
office buildings, windows, etc., all exhibit features that can
be recognised repeatably by regular polygon detection.
The literature related to this topic is vast, covering line
and analytical shape estimation and perceptual grouping,
as well as road sign detection. There are two alternative
schools of approach: model-driven approaches; and, perceptual grouping approaches that are frequently data driven.
The model-driven approaches typically employ retinotropic
mappings, whereby local features are detected purely locally, and pixel information is incorporated directly into
1
with a distance-based soft characteristic function.
m orienVariants of shape detector have also included
edge
p
g
tation information. The radial symmetry detector [13] improves on the circular Hough giving O(N"l) performance,
where l is the range of radii of the shapes being examined.
We have taken a similar approach to regular polygons [12].
However, only one regular polygon type could be detected
at a time and the angle vote applied to the whole image to
resolve orientation was computationally expensive.
Road sign recognition research has been around since
the mid 1980’s. Good results may be achieved with a classification approaches such as normalised cross correlation
2r tan(#/n)
[17]. However, this is computationally intensive per pixel.
A standard approach is to first apply a low computational
g
cost detection stage reducing classification to a fraction
of
"
the image stream. Typically this is through assumptions
r
(x,y)
about scene structure, colour, or a combination of both (e.g.,
[17, 9]). Although these assumptions are sound for many
road scenarios, the breakdown of a driver assistance system on hills and corners is not acceptable. Colour methods
also break down under the enormous variation in lighting
chrominance in outdoor situations, and segmenting out self
similar regions is not robust in scenes with dense overhead
branches that shadow the road. Instead, we propose a illumination invariant shape detection approach to find likely
candidates, to be followed by recognition.
We propose a computationally efficient algorithm to recover regular polygons using an a posteriori probability approach taking advantage of locality and gradient information. Rather than summing support, we recover the likelihood of the probability density function for all regular polygons in an image. The algorithm breaks the task into two
stages: 1) resolving likely regular polygons in a lower dimensional space; 2) taking the most likely regular polygons
and resolving their final parameters in higher dimensional
space. This allows us to recover all regular polygons in
one computationally efficient algorithm. The algorithm is
O(N rlw), where N is the size of the image, l is the maximum radius length, r is the number of radii being considered, and w is the width of the operation of the distance
function. w is close to one pixel for most implementations,
and r and l tend to be small numbers leading to a more efficient algorithm. We demonstrate that in constrained cases
this can lead to real-time implementations.
2
2a
r
(x,y)
(cx, cy)
!
!
Figure 1: A sample polygon five sides, with parameters as
defined above, and the line over which the centre may be for
a given oriented edge pixels, as described in Equation 10.
r
where the centre point
of the baseline falls on the circle.
The length of the individual sides is l = r2 tan nπ .
(c , cy)
We may define a xregular
polygon as having a centroid
in Euclidean 2-space! (cx , cy ), the radius r of the circle it
is circumscribed about, an orientation γ, and a number of
sides n. Thus we may define a regular polygon transform
as a function frp that maps the space of all possible regular polygons (regular polygon space) to 3D space. Regular
polygon space is five dimensional, and the transform forms
a mapping: frp : R4 , Z → R3 , where the integer dimension
is the number of sides, and to form closed polygons this is
three or greater, the space mapped to is image position plus
orientation of edges. Specifically, we label regular polygon
space: Φ = (cx , cy , r, γ, n). An example shape explaining
the parameters is shown in Figure 1.
Mapping all regular polygons into the image from 5D
regular polygon space is the integral across the space:
Z
frp =
(1)
The typical discrete image is a sampling over a regular
Cartesian grid. The gradient of the image is a set of gradient pixels x = (xj , j = 1, ...s) ∈ I corresponding to all
the edge pixels with orientation of the regular polygons (as
recovered by Sobel for example), plus any noise.
3
Likelihood formulation
A gradient image with a set of points x may be regarded
as the result of the transform from regular polygon space of
some set of polygons φm ∈ Φ, where m = 1 . . . M . In order to recover φm from the xj , we may estimate the probability density function over regular polygon space given xj ,
considering the image as the result of a mixture of regular
polygons, each of which is a Gaussian. That is:
Regular polygons
Archimedes gave a scientific method for calculating π to
arbitrary precision. The sum of the lengths of a regular
polygon of n sides inscribed on a circle is smaller than the
circumference, and the sum of the lengths of the sides circumscribed around a circle is greater than the perimeter of
the circle. We take our regular polygon definition that circumscribed around a circle. For an n sided regular polygon,
n
divide the circle into a set of n 2π
angle isosceles triangles,
frp (φ)dφ,
Φ
f=
M
X
αm p(φm |x),
(2)
m=1
where αm is a mixing parameter. If we assume a uniform
PM
mixture, then αm is constant for all m. As m=1 αm = 1,
we may drop this term, and apply Bayes Law:
M
X
m=1
p(φm |x) =
m
X
p(x|φm )p(φm )
p(x)
m=1
(3)
Note that there is a distinction between a regular polygon
in the scene and what appears in an image. An accidental
view where several straight lines at different scene depths
align to form a polygon is not a world structure, but is a
polygon in the image. We may say that any apparent regular polygon is a regular polygon in the image. Further,
with incomplete data, any edge pixel may be regarded as
part a regular polygon. Thus, the edge image can be seen
as being made up of regular polygons with noise. The result of this noise has two effects: edge displacement and
gradient error. Due to standard image noise, or an imperfect scene edge (e.g., a faded road sign), the gradient edge
in the image may be displaced slightly. There is also some
residual uncertainty of position due to image sampling. All
of these effects on the gradient edge may be reasonably approximated by a Gaussian in 3D over the parameters. The
gradient error results from the effect of intensity noise in
the image, and relation between orientation and intensity
values is a sinusoid. For small angles the error in orientation will be linear and therefore may also be approximated
by a Gaussian. Note that some authors choose to take gradient magnitude as a measure of edge certainty, however, it is
also highly correlated with the intensity contrast of the underlying image. Thus, we threshold the gradient magnitude
to basic noise, and set the remaining xj to unit magnitude.
If required, the magnitude can also be included as a continuous measure of gradient certainty in this formulation.
We may take it that the xj are the result of a set of polygons with missing data. Also, that the noise in the appearance of the edge pixels in their position and estimated orientation is additive and independent between the points. Then
we may take the probability of the Gaussian mixture of regular polygons in the image for the full set of points to be:
M
X
p(φm |x) =
m=1
m Y
X
p(xj |φm )p(φm )
p(xj )
m=1 j
(4)
Then, following Kiryati and Bruckstein [10], the probability density for an individual polygon φm and edge point
xj can be modelled as a Gaussian in 3D xj = (xj , yj , θj )
with zero mean and a point-specific covariance matrix:
P (xj |φm ) =
exp −g(xj , frp (Φ))Σ−1
j g(xj , frp (Φ)
1
2π|Σj | 2
,
(5)
where g(xj , frp (Φ)) is a function giving the distance from
the edge pixel to the closest point on the image projection
of the the regular polygon, Σj is the covariance matrix:
 2

σxj σxyj σxθj
Σj =  σxyj σy2j σyθj 
(6)
σxθj σyθj σθ2j
and |Σj | is the determinant of the covariance matrix. By
the independence of noise between points, we may take the
product over all edge points:
P (x|φm ) =
Y exp −g(xj , frp (Φ))Σ−1
j g(xj , frp (Φ)
1
2π|Σj | 2
j
,
(7)
Substituting back into Equation 3:
M
X
p(φm |x) =
(8)
m=1
m Y
X
exp(−g(xj ,frp (Φ))Σ−1
j g(xj ,frp (Φ))p(φm )
1
2π|Σj | 2 p(xj )
m=1 j
Taking log-likelihoods and collecting constants:
M
X
log(L(φm |x) =
m=1
m
X
m=1
X
(9)
−g(xj , frp (Φ))Σ−1
j g(xj , frp (Φ)) + const.
j
Matching to the regular polygon function: Two major
issues are posed by the formulation of Equation 9. The term
g(xj , frp (Φ)) is the distance from xj to the nearest point
on the regular polygon, i.e., this requires finding the nearest
point. Also, Φ is a 5D space, and so optimising may be
computationally expensive, while we would like to have a
real time implementation.
As regular polygon space is high dimensional, the image
is low dimensional, and edge points are sparse in general,
we may pose the problem as a mapping from image edge
pixels to possible regular polygons. We wish to form the
−1
from xj to regular polygons that it
inverse mapping frp
−1
will allow us to create a
may be a part of, i.e., Φ. frp
function g(xj , Φ) that gives a distance metric from the point
xj to the regular polygon defined by Φ.
−1
for a given xj , for unknown r and γ. If we
Consider frp
do not consider the edge gradient at xj , a regular polygon of
any number of sides may have a centre anywhere, the subspace of Φ that xj may be an element of is large, and the
−1
function frp
is intractable. However, we may recover orientation using a standard image processing operator (e.g.,
Sobel) giving xj = (xj , yj , θj ), where θj is the direction of
the maximum intensity gradient. In this case the orientation
of the regular polygon is constrained. Indeed, regardless
of n, (cx , cy ) is constrained to be on a line, orthogonal to
the gradient, at a distance r from (xj , yj ). This relation
is as shown in Figure 1. We maintain the standard of representing this in polar coordinates, but follow Weiman in
representing a straight-line segment in the complex plane
[22]. Consider that z1 is a complex number, with argument
π
2 − θj , and modulus 1, we have:
Z
l
t= 2r
h(xj ) =
xj + r(z1 t + iz1 )dt,
l
t=− 2r
(10)
where xj is the (xj , yj ) component of the point represented
in the complex plane, l is the side length given n and r.
Using Equation 10, we may define a new regular polygon
detection sub-space that is ambiguous in γ and n, but hence
only 3D, and computationally simple. To form the detection
subspace, we need only form the likelihood density function
of Equation 9 over (cx , cy , r). Let this new subspace be
Ψ = (cx , cy , r), and the function from this space to the
image be frps (s for subspace).
In order to collapse the dimension of number of sides,
we may assume the longest possible side length, that of a
triangle. In the case of other shapes with shorter sides, this
will lead to some edge pixels being considered that are not
actually part of the true regular polygon. However, these
ambiguities may be resolved subsequently. Thus, for any
particular ψ ∈ Ψ, we may define g as the minimum distance
from Ψ to the line defined in Equation 10.
Regular polygon detection: Using Equation 10 to find
likelihood density across a Ψ, we may take the peaks of the
distribution as the set of n regular polygons that are most
likely in this subspace, by the assumption of zero mean, if
we assume that the actual regular polygons are spatially separate. We will deal with the difficulties that arise from this
assumption subsequently. Rather than recover the variance
directly, we resolve the remaining dimensions in a second
stage. This separation is computationally motivated. By
finding most likely polygons in a reduced space we may focus computation on the most likely polygons, and ignoring
parts of Ψ1 that are not likely to have any regular polygons
present. This will be sufficient to recover all polygons to
some required likelihood, if we investigate regular polygons
in the reduced space down to the required likelihood as the
likelihood in Φ is an overestimate of that in Ψ.
In practice the variance constitutes: remaining dimensions of n and γ; and, regular polygons (or partially visible regular polygons) that are overlayed. This may result
in a figure that has a number edges of different length, for
which likelihood is overestimated. For example, consider
an octagon, where all sides are the length of sides for the
corresponding triangle, although this is a correct octagon,
its likelihood may be overestimated in Φ.
Resolving orientation and shape: Now we have resolved the mean for ψh = (cxh , cyh , rh ) ∈ Ψ for the set
of regular polygons that are most likely in Ψ. We may construct a likelihood function given this information to resolve
the remaining dimensions, that is:
may be assembled by maintaining a list of the parameters
for the xj that contributed to that point. This includes its
basic parameters, plus its distance as defined by g in the 3D
subspace, let this be dj . We may iterate over n, and integrate over γ and compute the log-likelihood of the probability density function over the remaining parameters using Equation 12. This is now a reduced dimensional space
search, however an efficient approximation to this algorithm
is discussed in the implementation.
4
Implementation
Computation of the complete regular polygon likelihood
distribution is expensive. Instead we discretely sample the
distribution in a manner that is appropriate for the application. Note that a coarse-to-fine strategy may be employed or
found peaks may be better located by subsequent EM. The
likelihood distribution can subsampled by a simple scan line
algorithm through xj . If one considers the distance function
to be the soft characteristic function of [14], then the loglikelihood estimation may be considered as a Hough-like
algorithm for estimation of regular polygon parameters.
Using g 2 , the likelihood contribution of a regular polygon of xj rapidly becomes small as distance increases.
Thus, if L(xj |Ψ) < , we set it to zero. Given this formulation, the algorithm can operate by, for each edge pixel, step
along the line of h for the width implied by , and adjust
the log likelihood of each sample point of the distribution
according to L(xj |Ψ). Alternatively, we may approximate
by adjusting only the closest pixels to the line, and convolving the likelihood with a Gaussian. Thus for each xj , we
update log-likelihoods for rlw points in Ψ, where w is the
width implied by . In practice, small w is sufficient, for a
fast implementation this may be one-two pixels. Also, for
most specific applications, r may be over a small range.
The second stage of the algorithm can also be implemented more computationally efficiently. In practice, one
is typically seeking either the set of k regular polygons with
the maximum likelihood, or all regular polygons with a likelihood greater than a threshold. Let us assume intially that
the xj in the list are all actually part of the polygon. In this
case, γ will be a mixture of Gaussians corresponding to the
orientations of each of the component edges. We may define
the log-likelihood of the probability of a regular polygon as
the sum over the edges, splitting Equation 12 over edges,
and replacing frp (Φ) with each of the edges from Equation
10. In order for an xj to be an element of any particular
log(L(o, n|x, cx , cy , r) =
(11) edge, θj must fall within the expected variance of the mean
X
for the orientation of the edge.
−g(xj , frp (Φ))Σ−1
j g(xj , frp (Φ)) + const
As edge length may be less with known n, an edge-based
j
distance function (with orientation) is more constrained
than g(xj , frps (Ψ)) adjusted to include orientation. Thus,
With a truncated distance function, for each point in deif we approximate the distance function, by its distance dj ,
tection subspace we may consider only the local xj , this
multiplied by a distance function over θj , this is an upper
1 and of the image if we truncate our distance function.
bound on the actual edge distance function. If we take the
Given edge elements xj of an image, detect regular polygons with n ∈ N sides. For each polygon radius r:
1. Estimate likelihood image:
(a) For each xj : Compute all triangle locations pk
that could generate xj , accumulate locations in
a vote image, and record information: pk , ∠xj
and distance mk along line h.
(b) Convolve vote image with Gaussian to generate
discrete approximation of likelihood image.
2. Evaluate each maxima qi for each n ∈ N :
(a) Build a weighted angular histogram of all
recorded ∠xj values with mk < l(n,r)
,
2
weighted by a Gaussian of kpk − qi k2 .
(b) Convolve histogram with string of delta functions δ(γ − 2π
) corresponding with the p peak
n
edges. The γ that maximises this convolution is
the most likely orientation of an n sided polygon, and the values at γ ± 2π
u, u ∈ Z indicate
n
the support for each side.
(c) Total support is determined as a function of the
support for each side of the shape.
Figure 2: Summary of the algorithm.
weighted xj over θ and convolve it with a Gaussian corresponding to our expected variance in θ, then this forms
an approximate upper bound on the log-likelihood of the
probability density function of individual edge orientations.
We may use this to hypothesis test for γ and n, by taking
each hypothesis in turn and summing over edges in the upper bound density function by direct look up at the expected
angles, to form the upper bound density function across our
remaining parameters. Rather than for all γ we may take
the top p peaks as this corresponds to upper bounds likelihoods of the most likely edges. For each of these we may
form the complete likelihood function of Equation 12. As
the mean of the maximum likelihood orientation may not
correspond to the mean orientation of the most likely edge.
If required an EM search may be performed over orientation
in the neighbourhood of these peaks.
For real applications, the number of radii being explored
is not large, and the sampling can be coarse. Further if it is
acceptable to overstate the likelihood, and precise position
is not required, the algorithm may cease without removing
edge pixels. Implementation is summarised in Figure 2.
With no a priori scene information, we may assume that
the xj appear with an isotropic distribution, except for regular polygons. However, in a road scene, for example, the
edges of the road and dividing lines often result in long
piecewise straight edges. In this case, the probability of
multiple edge points appearing together, with the same gradient, that are not part of something we wish to consider as
a regular polygon, is greater than the probability of pixels
appearing together randomly in the image.
In such a case, we may determine a priori distributions
from a set of images that is representative of the incoming
images. From this we may adjust the likelihoods to reflect
the number of supporting edges from the support for each
of the edges. In such long clear line scenarios, we found it
was effective to incorporate negative probability weighting
at the ends of the line of influence of the xj . This prevented
over emphasis of strongly contrasting long lines.
Finally, where other information is known a priori we
may incorporate this information into our function, and simplify computation. This will be explained in more detail in
the context of road sign detection in the results.
5
Experimental results
We evaluate performance in the presence of noise in artificial images, as well as on real images. By incorporating
prior knowledge about the appearance of the road scene, we
were able to adapt the algorithm to run at 16Hz for 320×240
images in which it was able to reliably detect a road sign in a
test sequence taken from the vehicle in Figure 5(a). Finally,
we demonstrate the application of a constrained version as
a feature detector for wide-baseline matching for robot corridor navigation on the robot shown in Figure 5(b).
On still images: The detector was evaluated on a range
of images containing regular polygons. Figures 3 and 4
show detection results for searching for 3 to 8 sided regular polygons over radii r ∈ {8, 11, ...17}. Note that this
non-continuous range of radii is adequate to detect shapes
at neighbouring radii. Figure 3(b) shows the approximation
of the likelihood image generated by the algorithm; note the
peaks at the centres of the shapes detected in (c). The final
output (d) illustrates both the impressive detection performance, and illustrates some of the artefacts of the method.
Owing to the Gaussian modelling of edge and centroid locations the shapes detected do not always exactly overlay the
edge locations of the original shapes. Also, polygons may
be found with alignment of partial edges, and are awarded
low (but not zero) strength accordingly.
The majority of incorrect hypotheses correspond to polygons of a small number of sides. There are two reasons for
this: firstly, (for n ≤ 8) as n increases it becomes less likely
that n edges will be ‘accidentally’ aligned by noise; secondly the prevalence of right angles in built environments
gives a strong prior for finding squares.
Figure 4 shows the detector operating on outdoor images, here the near-perfect regular polygon nature of the
road signs ensure that they are strongly detected. The top
right figure shows an example of the detected square shapes
common around man-made structures, however, for sign detection the orientation of the diamond-shaped warning sign
in the foreground sets it apart from the other squares.
Robustness to noise: The 180 × 240 test image I :
I(p) ∈ [0.25, 0.75] in Figure 3 (e) was corrupted by increasing amounts of additive Gaussian noise σ = 0, 0.05, ..., 1
and performance was evaluated over 20 runs at each noise
(a)
(b)
(e)
(f)
(c)
(d)
(g)
Figure 3: (a) Input image, (b) likelihood image for r = 17, (c) results for r = 17, and (d) results across all radii. Detection
results fade from red (strong) to blue (weak). (e) Noise test image, (f) with SNR=3Db (σ = 0.5) (g) mean and standard
deviation of detection performance with additive Gaussian noise.
(a)
Figure 4: Performance on outdoor images.
level at radius r = 20. Performance was quantified from
the top six shape hypotheses which were taken to be correct if they located the shape with the correct number of
sides within ±4 pixels of its ground truth location. Figure 3
(f) shows the average detection performance versus noise,
demonstrating stability in the presence of noise. At σ = 0.5
the average detection rate drops below 50%, but at this point
it is difficult even for a person to differentiate between some
of the shapes, Figure 3 (e).
Real-time sign detection: Critical information signs
with information that requires driver action regularly appear
on triangular, diamond, or octagonal backgrounds. In road
scenes signs are designed to be easily visible; they appear
at set orientation (approximately) unless damaged. Further,
if the camera axis aligns with the vehicle’s forward motion,
(b)
Figure 5: (a) Inside the intelligent vehicle. Cameras monitor the road scene appear in place of the rear-vision mirror.
(b) The NOMAD robot for navigation in corridor-like environments with a camera pair on top.
signs will always be parallel to the image plane when they
are close, even on curved roads and will be visible for many
frames. Also, for a camera of approximately known focal length, their apparent size (radius) is constrained over
a narrow range. Signs smaller than a few pixels cannot be
recognised, so need not be detected. As signs have maximal
world size, physical constraints mean they will never appear
more closely than a set distance from the vehicle.
Detection aim to reduce each image to a small number
of possible candidates. A sign need not be detected in every frame, but must be reliably detected while it is visible.
Due to the appearance of regular polygons due to accidental
features, it is efficacious to require a regular polygon to be
detected over several images at a similar position and size as
such accidental views will occur frequently in robot vision,
but often will not be sustained.
In this trial, we constrained the detector to giveway and
roundabout signs, triangles, where the top edge is parallel to
the ground plane, thus the number of sides is constrained,
and orientation is constrained to a range a priori. For the
standard configuration of the vehicle used in this sequence,
four separate radii were sufficient of 6, 8, 10, and 12 pixels.
The a priori constraint over number of sides and radii meant
that gradient orientations outside the constraints could not
be part of these oriented polygons, so could be disregarded
a priori. These orientations were 0, 120 and 240 ◦ , each
±12◦ . Thus, only the first stage of the algorithm was required to recover all ambiguity. The constrained algorithm
was able to run in less than 50ms per frame (320x240 image), based on an implementation in C++ on a standard PC,
equivalent to that mounted in the vehicle. This is quite adequate for a sign to be visible for many frames in any reasonable situation for a road vehicle, and hence is real-time.
The road trial is on a sequence, see Figure 6. Of the
sequence the sign is of a detectable size for 48 frames. A
sample detection image is shown in Figure 6. For detection,
we required the sign to be one of the two most likely triangles for any radius for an initial image, and then one of
the top 10 for the next k images. False positive and false
negative curves are shown in Figure 7. Although the sign
was not detected as one of the top two candidates in every
image, it was reliably detected many times over the image
sequence. The processing of recognition using normalised
cross correlation for each candidate is less than 1ms in our
system, so the number of false positives was well within
computation limits for real time processing.
Indoor scenes: In a project to develop vision-based
robotic mapping, we applied the regular polygon detector
to detect square-like features. Here n is set, but γ must be
resolved, i.e., 1D ambiguity after initial detection. The detector was run at three different radii, and found all squares
above a low likelihood (a few pixels on three edges). It detects stable edges that are the basis of squares, but is tolerant
to incomplete edges and affine distortion. Figure 8 shows a
sample image. The detector was applied to pair of images
taken from the robot as it was navigating around indoor office environment, moving at approximately 30 cm/sec. With
images three seconds apart the baseline was wide, approximately one metre. On corners the robot turned mostly in
Figure 6: An image from the middle of the roundabout sign
sequence, with most likely triangles backprojected, and the
likelihood function for radius 8.
Figure 7: Average and standard deviation of number of false
positives (top) and false negatives (bottom), given number
of frames k that the sign is required to be present.
place, leading to shifts of more than half the image of features, see Figure 9. Features were matched according to
size and orientation, the local greyscale environment and
position, and used to calculate the fundamental matrix to
recover motion. Results showed a high percentage of the
found points matched (around 70% typically) correctly over
frames moving down the corridor, but reduced over rotation
as expected. Figure 9 shows a typical images taken from
the sequence of 69 images as it turns around a corner. The
matches show promising initial results. Figure 9 (c) shows
the matches obtained by SIFT [11], using the reference implementation with its packaged parameters.2 Note that the
square feature detector will be most effective in this type of
rectilinear environment, however, it is highly suitable to be
part of a battery of detectors for matching. Its speed of operation makes it plausible for robotic mapping applications.
6
Conclusions
Regular polygon detection is an important problem with
multiple applications in robotics and computer vision. Using an aposteriori probability approach, we presented a new
2 http://www.cs.ubc.ca/
lowe/keypoints/
Figure 8: Square features: size and orientation are represented directly, likelihood is represented by colour, where
red is most certain, and yellow is least certain.
[9] S-H Hsu and C-L Huang. Road sign detection and recognition
using matching pursuit method. Image and Vision Computing,
19:119–129, 2001.
[10] N Kiryati and A M Bruckstein. Heteroscedastic hough transform (htht): An efficient method for robust line fitting in the
‘errors in the variables’ problem. Computer Vision and Image
Understanding, 78(1):69–83, Apr. 2000.
(a)
(b)
(c)
Figure 9: Matches for turning a corner. The pink lines represent correct matches, while the others represent erroneous
matches. (c) Matches found using SIFT, only one found.
algorithm for detecting regular polygons. We defined the
continuous log-likelihood of the probability density function of regular polygons. In order to make the algorithm
computationally efficient we find initial likely regular polygons in a lower dimensional space, and then resolve the remaining parameters for likely regular polygons. We presented an efficient algorithm based on this, and adaptations
of this algorithm that can run at 16Hz detecting signs in an
intelligent vehicle application. Experimental results show
the efficacy and robustness of the algorithm, and its application to driver assistance, and as a feature detector for
matching, applied to a robot sequence for mapping.
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